In two dimensional geometry, the set of points that are equidistant from two given points is the perpendicular bisector of the line segment between them- a line.
Do you see how the same reasoning in three dimensions gives the plane passes through the midpoint of the line segment between the two points and is perpendicular to it?
Analytically, if one point is (a, b, c) and the other is (u, v, w), saying that (x, y, z) is equidistant from them means that .
Square both sides to get rid of the square roots: .
Do all of those squares and you will see that the , , and terms cancel, leaving only a linear equation.